Exceptional Jordan Algebra Research Articles

On Jordan algebras that are factors of Matsuo algebras

We describe all finite connected 3-transposition groups whose Matsuo algebras have nontrivial factors that are Jordan algebras. As a corollary, we show that if $\mathbb{F}$ is a field of characteristic 0, then there exist infinitely many primitive axial algebras of Jordan type $\frac{1}{2}$ over $\mathbb{F}$ that are not factors of Matsuo algebras. As an example, we prove this for an exceptional Jordan algebra over~$\mathbb{F}$.

Octonionic Magical Supergravity, Niemeier Lattices, and Exceptional & Hilbert Modular Forms

AbstractThe quantum degeneracies of Bogomolny‐Prasad‐Sommerfield (BPS) black holes of octonionic magical supergravity in five dimensions are studied. Quantum degeneracy is defined purely number theoretically as the number of distinct states in charge space with a given set of invariant labels. Quantum degeneracies of spherically symmetric stationary BPS black holes are given by the Fourier coefficients of modular forms of exceptional group . Their charges take values in the lattice defined by the exceptional Jordan algebra over integral octonions. The quantum degeneracies of rank 1 and rank 2 BPS black holes are given by the Fourier coefficients of singular modular forms and . The rank 3 (large) BPS black holes will be studied elsewhere. Following the work of N. Elkies and B. Gross on embeddings of cubic rings A into the exceptional Jordan algebra we show that the quantum degeneracies of rank 1 black holes described by such embeddings are given by the Fourier coefficients of the Hilbert modular forms (HMFs) of . If the discriminant of the cubic ring A is with p a prime number then the isotropic lines in the 24 dimensional orthogonal complement of A define a pair of Niemeier lattices which can be taken as charge lattices of some BPS black holes. The current status of the searches for the M/superstring theoretic origins of the octonionic magical supergravity is also reviewed.

Gravitation, and quantum theory, as emergent phenomena

There must exist a reformulation of quantum field theory, even at low energies, which does not depend on classical time. The octonionic theory proposes such a reformulation, leading to a pre-quantum pre-spacetime theory. The ingredients for constructing such a theory, which is also a unification of the standard model with gravitation, are : (i) the pre-quantum theory of trace dynamics – a matrix-valued Lagrangian dynamics, (ii) the spectral action principle of non-commutative geometry, (iii) the number system known as the octonions, for constructing a non-commutative manifold and for defining elementary particles via Clifford algebras, (iv) a Lagrangian with E 8 × E 8 symmetry. The split bioctonions define a sixteen dimensional space (with left-right symmetry) whose geometry (evolving in Connes time) relates to the four known fundamental forces, while predicting two new forces, SU(3) grav and U(1) grav . This latter interaction is possibly the theoretical origin of MOND. Coupling constants of the standard model result from left-right symmetry breaking, and their values are theoretically determined by the characteristic equation of the exceptional Jordan algebra of the octonions. The quantum-to-classical transition, precipitated by the entanglement of a critical number of fermions, is responsible for the emergence of classical spacetime, and also for the familiar formulation of quantum theory on a spacetime background.

Maximal antipodal sets of E6 and some compact symmetric spaces

We explicitly classify congruent classes of maximal antipodal sets of E6 and some compact symmetric spaces related to E6. Moreover, we give realizations of these compact symmetric spaces by using some subalgebras of the complex exceptional Jordan algebra JC. In this realization, we explicitly describe congruent classes of maximal antipodal sets of these compact symmetric spaces.

Quantum gravity effects in the infrared: a theoretical derivation of the low-energy fine structure constant and mass ratios of elementary particles

We have recently proposed a pre-quantum, pre-spacetime theory as a matrix-valued Lagrangian dynamics on an octonionic spacetime. This theory offers the prospect of unifying internal symmetries of the standard model with pre-gravitation. We explain why such a quantum gravitational dynamics is in principle essential even at energies much smaller than Planck scale. The dynamics can also predict the values of free parameters of the low-energy standard model: these parameters arising in the Lagrangian are related to the algebra of the octonions, which define the underlying non-commutative spacetime on which the dynamical degrees of freedom evolve. These free parameters are related to the exceptional Jordan algebra \(J_3(8)\), which describes the three fermion generations. We use the octonionic representation of fermions to compute the eigenvalues of the characteristic equation of this algebra and compare the resulting eigenvalues with known mass ratios for quarks and leptons. We show that the ratios of the eigenvalues correctly reproduce the [square root of the] known mass ratios. In conjunction with the trace dynamics Lagrangian, these eigenvalues also yield a theoretical derivation of the low-energy fine structure constant.

Majorana neutrinos, exceptional Jordan algebra, and mass ratios for charged fermions

Abstract We provide theoretical evidence that the neutrino is a Majorana fermion. This evidence comes from assuming that the standard model and beyond-standard-model physics can be described through division algebras, coupled to quantum dynamics. We use the division algebras scheme to derive mass ratios for the standard model charged fermions of three generations. The predicted ratios agree well with the observed values if the neutrino is assumed to be Majorana. However, the theoretically calculated ratios completely disagree with known values if the neutrino is taken to be a Dirac particle. Towards the end of the article we discuss prospects for unification of the standard model with gravitation if the assumed symmetry group of the theory is E 6, and if it is assumed that space-time is an 8D octonionic space-time, with 4D Minkowski space-time being an emergent approximation. Remarkably, we find evidence that the precursor of classical gravitation, described by the symmetry SU(3)grav × SU(2)R × U(1)grav is the right-handed counterpart of the standard model SU(3)color × SU(2)L × U(1) Y . This provides the theoretical justification for the mass-ratios analysis based on the eigenvalues of the exceptional Jordan algebra.

Octonionic Planes and Real Forms of $G_2$, $F_4$ and $E_6$

In this work we present a useful way to introduce the octonionic projective and hyperbolic plane $\mathbb{O}P^{2}$ through the use of Veronese vectors. Then we focus on their relation with the exceptional Jordan algebra $\mathfrak{J}_{3}^{\mathbb{O}}$ and show that the Veronese vectors are the rank-one elements of the algebra. We then study groups of motions over the octonionic plane recovering all real forms of $\text{G}_{2}$, $\text{F}_{4}$ and $\text{E}_{6}$ groups and finally give a classification of all octonionic and split-octonionic planes as symmetric spaces.

Complexification of the Exceptional Jordan Algebra and Its Application to Particle Physics

Recent papers contributed revitalizing the study of the exceptional Jordan algebra $\mathfrak{h}_{3}(\mathbb{O})$ in its relations with the true Standard Model gauge group $\mathrm{G}_{SM}$. The absence of complex representations of $\mathrm{F}_{4}$ does not allow $\Aut\left(\mathfrak{h}_{3}(\mathbb{O})\right)$ to be a candidate for any Grand Unified Theories, but the automorphisms of the complexification of this algebra, i.e., $\mathfrak{h}_{3}^{\mathbb{C}}(\mathbb{O})$, are isomorphic to the compact form of $\mathrm{E}_{6}$ and similar constructions lead to the gauge group of the minimal left-right symmetric extension of the Standard Model.

Quantum theory without classical time: Octonions, and a theoretical derivation of the fine structure constant 1/137

There must exist a reformulation of quantum field theory which does not refer to classical time. We propose a pre-quantum, pre-spacetime theory, which is a matrix-valued Lagrangian dynamics for gravity, Yang–Mills fields, and fermions. The definition of spin in this theory leads us to an eight-dimensional octonionic spacetime. The algebra of the octonions reveals the standard model; model parameters are determined by roots of the cubic characteristic equation of the exceptional Jordan algebra. We derive the asymptotic low-energy value 1/137 of the fine structure constant, and predict the existence of universally interacting spin one Lorentz bosons, which replace the hypothesised graviton. Gravity is not to be quantized, but is an emergent four-dimensional classical phenomenon, precipitated by the spontaneous localisation of highly entangled fermions.

SO(9) characterization of the standard model gauge group

A recent series of works characterized the Standard Model (SM) gauge group GSM as the subgroup of SO(9) that, in the octonionic model of the latter, preserves the split O=C⊕C3 of the space of octonions into a copy of the complex plane plus the rest. This description, however, proceeded via the exceptional Jordan algebras J3(O),J2(O) and, in this sense, remained indirect. One of the goals of this paper is to provide as explicit a description as possible and also to clarify the underlying geometry. The other goal is to emphasize the role played by different complex structures in the spaces O and O2. We provide a new characterization of GSM: The group GSM is the subgroup of Spin(9) that commutes with a certain complex structure JR in the space O2 of Spin(9) spinors. The complex structure JR is parameterized by a choice of a unit imaginary octonion. This characterization of GSM is essentially octonionic in the sense that JR is restrictive because octonions are non-associative. The quaternionic analog of JR is the complex structure in the space H2 of Spin(5) spinors that commutes with all Spin(5) transformations.

Composites and Categories of Euclidean Jordan Algebras

We consider possible non-signaling composites of probabilistic models based on euclidean Jordan algebras (EJAs), satisfying some reasonable additional constraints motivated by the desire to construct dagger-compact categories of such models. We show that no such composite has the exceptional Jordan algebra as a direct summand, nor does any such composite exist if one factor has an exceptional summand, unless the other factor is a direct sum of one-dimensional Jordan algebras (representing essentially a classical system). Moreover, we show that any composite of simple, non-exceptional EJAs is a direct summand of their universal tensor product, sharply limiting the possibilities.These results warrant our focussing on concrete Jordan algebras of hermitian matrices, i.e., euclidean Jordan algebras with a preferred embedding in a complex matrix algebra. We show that these can be organized in a natural way as a symmetric monoidal category, albeit one that is not compact closed. We then construct a related category InvQM of embedded euclidean Jordan algebras, having fewer objects but more morphisms, that is not only compact closed but dagger-compact. This category unifies finite-dimensional real, complex and quaternionic mixed-state quantum mechanics, except that the composite of two complex quantum systems comes with an extra classical bit.Our notion of composite requires neither tomographic locality, nor preservation of purity under tensor product. The categories we construct include examples in which both of these conditions fail. {In such cases, the information capacity (the maximum number of mutually distinguishable states) of a composite is greater than the product of the capacities of its constituents.}

Trace dynamics and division algebras: towards quantum gravity and unification

Abstract We have recently proposed a Lagrangian in trace dynamics at the Planck scale, for unification of gravitation, Yang–Mills fields, and fermions. Dynamical variables are described by odd-grade (fermionic) and even-grade (bosonic) Grassmann matrices. Evolution takes place in Connes time. At energies much lower than Planck scale, trace dynamics reduces to quantum field theory. In the present paper, we explain that the correct understanding of spin requires us to formulate the theory in 8-D octonionic space. The automorphisms of the octonion algebra, which belong to the smallest exceptional Lie group G 2, replace space-time diffeomorphisms and internal gauge transformations, bringing them under a common unified fold. Building on earlier work by other researchers on division algebras, we propose the Lorentz-weak unification at the Planck scale, the symmetry group being the stabiliser group of the quaternions inside the octonions. This is one of the two maximal sub-groups of G 2, the other one being SU(3), the element preserver group of octonions. This latter group, coupled with U(1) em , describes the electrocolour symmetry, as shown earlier by Furey. We predict a new massless spin one boson (the ‘Lorentz’ boson) which should be looked for in experiments. Our Lagrangian correctly describes three fermion generations, through three copies of the group G 2, embedded in the exceptional Lie group F 4. This is the unification group for the four fundamental interactions, and it also happens to be the automorphism group of the exceptional Jordan algebra. Gravitation is shown to be an emergent classical phenomenon. Although at the Planck scale, there is present a quantised version of the Lorentz symmetry, mediated by the Lorentz boson, we argue that at sub-Planck scales, the self-adjoint part of the octonionic trace dynamics bears a relationship with string theory in 11 dimensions.

A note on Jordan algebras, three generations and exceptional periodicity

It is shown that the algebra [Formula: see text] based on the complexified Exceptional Jordan, and the complex Clifford algebra in 4D, is rich enough to describe all the spinorial degrees of freedom of three generations of fermions in 4D, and include additional fermionic dark matter candidates. Furthermore, the model described in this paper can account also for the Standard Model gauge symmetries. We extend these results to the Magic Star algebras of Exceptional Periodicity developed by Marrani–Rios–Truini and based on the Vinberg cubic [Formula: see text] algebras which are generalizations of exceptional Jordan algebras. It is found that there is a one-to-one correspondence among the real spinorial degrees of freedom of four generations of fermions in 4D with the off-diagonal entries of the spinorial elements of the [Formula: see text] [Formula: see text] of Vinberg matrices at level [Formula: see text]. These results can be generalized to higher levels [Formula: see text] leading to a higher number of generations beyond 4. Three [Formula: see text] of [Formula: see text] algebras and their conjugates [Formula: see text] were essential in the Magic Star construction of Exceptional Periodicity that extends the [Formula: see text] algebra to [Formula: see text] with [Formula: see text] integer.

Albert algebras over rings and related torsors

AbstractWe study exceptional Jordan algebras and related exceptional group schemes over commutative rings from a geometric point of view, using appropriate torsors to parametrize and explain classical and new constructions, and proving that over rings, they give rise to nonisomorphic structures.We begin by showing that isotopes of Albert algebras are obtained as twists by a certain $\mathrm F_4$ -torsor with total space a group of type $\mathrm E_6$ and, using this, that Albert algebras over rings in general admit nonisomorphic isotopes even in the split case, as opposed to the situation over fields. We then consider certain $\mathrm D_4$ -torsors constructed from reduced Albert algebras, and show how these give rise to a class of generalised reduced Albert algebras constructed from compositions of quadratic forms. Showing that this torsor is nontrivial, we conclude that the Albert algebra does not uniquely determine the underlying composition, even in the split case. In a similar vein, we show that a given reduced Albert algebra can admit two coordinate algebras which are nonisomorphic and have nonisometric quadratic forms, contrary, in a strong sense, to the case over fields, established by Albert and Jacobson.

A simple and quantum-mechanically motivated characterization of the formally real Jordan algebras

Quantum theory’s Hilbert space apparatus in its finite-dimensional version is nearly reconstructed from four simple and quantum-mechanically motivated postulates for a quantum logic. The recon- struction process is not complete, since it excludes the two-dimensional Hilbert space and still includes the exceptional Jordan algebras, which are not part of the Hilbert space apparatus. Options for physically meaningful potential generalizations of the apparatus are discussed.

Magic Star and Exceptional Periodicity: an approach to Quantum Gravity

We present a periodic infinite chain of finite generalisations of the exceptional structures, including the exceptional Lie algebra e8, the exceptional Jordan algebra (and pair) and the octonions. We will also argue on the nature of space-time and indicate how these algebraic structures may inspire a new way of going beyond the current knowledge of fundamental physics.

Constructing numbers in quantum gravity: infinions

Based on the Cayley-Dickson process, a sequence of multidimensional structured natural numbers (infinions) creates a path from quantum information to quantum gravity. Octonionic structure, exceptional Jordan algebra, and Lie algebra are encoded on a graph with E9 connectivity, decorated by integral matrices. With the magic star, a toy model for a quantum gravity is presented with its naturally emergent quasicrystalline projective compactification.

Exceptional quantum geometry and particle physics II

We continue the study undertaken in [13] of the relevance of the exceptional Jordan algebra J38 of hermitian 3×3 octonionic matrices for the description of the internal space of the fundamental fermions of the Standard Model with 3 generations. By using the suggestion of [30] (properly justified here) that the Jordan algebra J28 of hermitian 2×2 octonionic matrices is relevant for the description of the internal space of the fundamental fermions of one generation, we show that, based on the same principles and the same framework as in [13], there is a way to describe the internal space of the 3 generations which avoids the introduction of new fundamental fermions and where there is no problem with respect to the electroweak symmetry.

Octonions, Exceptional Jordan Algebra and The Role of The Group $$F_4$$ F 4 in Particle Physics

Normed division rings are reviewed in the more general framework of composition algebras that include the split (indefinite metric) case. The Jordan - von Neumann - Wigner classification of finite dimensional Jordan algebras is outlined with special attention to the 27 dimensional exceptional Jordan algebra J. The automorphism group F_4 of J and its maximal Borel - de Siebenthal subgroups are studied in detail and applied to the classification of fundamental fermions and gauge bosons. Their intersection in F_4 is demonstrated to coincide with the gauge group of the Standard Model of particle physics. The first generation's fundamental fermions form a basis of primitive idempotents in the euclidean extension of the Jordan subalgebra JSpin_9 of J.

Deducing the symmetry of the standard model from the automorphism and structure groups of the exceptional Jordan algebra

We continue the study undertaken in Ref. 16 of the exceptional Jordan algebra [Formula: see text] as (part of) the finite-dimensional quantum algebra in an almost classical space–time approach to particle physics. Along with reviewing known properties of [Formula: see text] and of the associated exceptional Lie groups we argue that the symmetry of the model can be deduced from the Borel–de Siebenthal theory of maximal connected subgroups of simple compact Lie groups.